Trigonometry Examples

$\cos (\frac{11 π}{12}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1

The exact value of $\cos (\frac{11 π}{12})$ is $- \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \cos (\frac{π}{12}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.2

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$- \cos (\frac{π}{4} - \frac{π}{6}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.3

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$- (\cos (\frac{π}{4}) \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.4

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.5

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (\frac{π}{4}) \sin (\frac{π}{6})) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.6

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.7

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.8

Simplify $- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$- (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.8.1.1.2

Combine using the product rule for radicals.

$- (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.8.1.1.3

Multiply $2$ by $3$ .

$- (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.8.1.1.4

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.8.1.2

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.8.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.8.1.2.2

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 1.8.2

Combine the numerators over the common denominator.

$- \frac{\sqrt{6} + \sqrt{2}}{4} \cos (\frac{3 π}{4}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \frac{\sqrt{6} + \sqrt{2}}{4} (- \cos (\frac{π}{4})) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{6} + \sqrt{2}}{4} (- \frac{\sqrt{2}}{2}) + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 4

Multiply $- \frac{\sqrt{6} + \sqrt{2}}{4} (- \frac{\sqrt{2}}{2})$ .

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Step 4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{6} + \sqrt{2}}{4} \frac{\sqrt{2}}{2} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 4.2

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4}$ by $1$ .

$\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 4.3

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{(\sqrt{6} + \sqrt{2}) \sqrt{2}}{4 \cdot 2} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 4.4

Multiply $4$ by $2$ .

$\frac{(\sqrt{6} + \sqrt{2}) \sqrt{2}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 5

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{2}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 6

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 2} + \sqrt{2} \sqrt{2}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 7

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 2} + \sqrt{2 \cdot 2}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 8

Simplify each term.

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Step 8.1

Multiply $6$ by $2$ .

$\frac{\sqrt{12} + \sqrt{2 \cdot 2}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 8.2

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.2.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} + \sqrt{2 \cdot 2}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 8.2.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 3} + \sqrt{2 \cdot 2}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 8.3

Pull terms out from under the radical.

$\frac{2 \sqrt{3} + \sqrt{2 \cdot 2}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 8.4

Multiply $2$ by $2$ .

$\frac{2 \sqrt{3} + \sqrt{4}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 8.5

Rewrite $4$ as $2^{2}$ .

$\frac{2 \sqrt{3} + \sqrt{2^{2}}}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 8.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2 \sqrt{3} + 2}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 9

Cancel the common factor of $2 \sqrt{3} + 2$ and $8$ .

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Step 9.1

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{2 (\sqrt{3}) + 2}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 9.2

Factor $2$ out of $2$ .

$\frac{2 (\sqrt{3}) + 2 (1)}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 9.3

Factor $2$ out of $2 (\sqrt{3}) + 2 (1)$ .

$\frac{2 (\sqrt{3} + 1)}{8} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $2$ out of $8$ .

$\frac{2 (\sqrt{3} + 1)}{2 (4)} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 9.4.2

Cancel the common factor.

$\frac{2 (\sqrt{3} + 1)}{2 \cdot 4} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 9.4.3

Rewrite the expression.

$\frac{\sqrt{3} + 1}{4} + \sin (\frac{11 π}{12}) \sin (\frac{3 π}{4})$

Step 10

The exact value of $\sin (\frac{11 π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 10.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sqrt{3} + 1}{4} + \sin (\frac{π}{12}) \sin (\frac{3 π}{4})$

Step 10.2

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\frac{\sqrt{3} + 1}{4} + \sin (\frac{π}{4} - \frac{π}{6}) \sin (\frac{3 π}{4})$

Step 10.3

Apply the difference of angles identity.

$\frac{\sqrt{3} + 1}{4} + (\sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \sin (\frac{3 π}{4})$

Step 10.4

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \sin (\frac{3 π}{4})$

Step 10.5

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6})) \sin (\frac{3 π}{4})$

Step 10.6

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) \sin (\frac{3 π}{4})$

Step 10.7

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (\frac{3 π}{4})$

Step 10.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 10.8.1

Simplify each term.

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Step 10.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 10.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (\frac{3 π}{4})$

Step 10.8.1.1.2

Combine using the product rule for radicals.

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (\frac{3 π}{4})$

Step 10.8.1.1.3

Multiply $2$ by $3$ .

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (\frac{3 π}{4})$

Step 10.8.1.1.4

Multiply $2$ by $2$ .

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (\frac{3 π}{4})$

Step 10.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 10.8.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \sin (\frac{3 π}{4})$

Step 10.8.1.2.2

Multiply $2$ by $2$ .

$\frac{\sqrt{3} + 1}{4} + (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \sin (\frac{3 π}{4})$

Step 10.8.2

Combine the numerators over the common denominator.

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{6} - \sqrt{2}}{4} \sin (\frac{3 π}{4})$

Step 11

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{6} - \sqrt{2}}{4} \sin (\frac{π}{4})$

Step 12

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2}$

Step 13

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2}$ .

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Step 13.1

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{3} + 1}{4} + \frac{(\sqrt{6} - \sqrt{2}) \sqrt{2}}{4 \cdot 2}$

Step 13.2

Multiply $4$ by $2$ .

$\frac{\sqrt{3} + 1}{4} + \frac{(\sqrt{6} - \sqrt{2}) \sqrt{2}}{8}$

Step 14

Apply the distributive property.

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{2}}{8}$

Step 15

Combine using the product rule for radicals.

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{6 \cdot 2} - \sqrt{2} \sqrt{2}}{8}$

Step 16

Multiply $- \sqrt{2} \sqrt{2}$ .

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Step 16.1

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{6 \cdot 2} - ({\sqrt{2}}^{1} \sqrt{2})}{8}$

Step 16.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{6 \cdot 2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{8}$

Step 16.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{6 \cdot 2} - {\sqrt{2}}^{1 + 1}}{8}$

Step 16.4

Add $1$ and $1$ .

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{6 \cdot 2} - {\sqrt{2}}^{2}}{8}$

Step 17

Simplify each term.

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Step 17.1

Multiply $6$ by $2$ .

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{12} - {\sqrt{2}}^{2}}{8}$

Step 17.2

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 17.2.1

Factor $4$ out of $12$ .

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{4 (3)} - {\sqrt{2}}^{2}}{8}$

Step 17.2.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{2^{2} \cdot 3} - {\sqrt{2}}^{2}}{8}$

Step 17.3

Pull terms out from under the radical.

$\frac{\sqrt{3} + 1}{4} + \frac{2 \sqrt{3} - {\sqrt{2}}^{2}}{8}$

Step 17.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 17.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{3} + 1}{4} + \frac{2 \sqrt{3} - {(2^{\frac{1}{2}})}^{2}}{8}$

Step 17.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{3} + 1}{4} + \frac{2 \sqrt{3} - 2^{\frac{1}{2} \cdot 2}}{8}$

Step 17.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{3} + 1}{4} + \frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8}$

Step 17.4.4

Cancel the common factor of $2$ .

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Step 17.4.4.1

Cancel the common factor.

$\frac{\sqrt{3} + 1}{4} + \frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8}$

Step 17.4.4.2

Rewrite the expression.

$\frac{\sqrt{3} + 1}{4} + \frac{2 \sqrt{3} - 2^{1}}{8}$

Step 17.4.5

Evaluate the exponent.

$\frac{\sqrt{3} + 1}{4} + \frac{2 \sqrt{3} - 1 \cdot 2}{8}$

Step 17.5

Multiply $- 1$ by $2$ .

$\frac{\sqrt{3} + 1}{4} + \frac{2 \sqrt{3} - 2}{8}$

Step 18

Cancel the common factor of $2 \sqrt{3} - 2$ and $8$ .

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Step 18.1

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{\sqrt{3} + 1}{4} + \frac{2 (\sqrt{3}) - 2}{8}$

Step 18.2

Factor $2$ out of $- 2$ .

$\frac{\sqrt{3} + 1}{4} + \frac{2 (\sqrt{3}) + 2 (- 1)}{8}$

Step 18.3

Factor $2$ out of $2 (\sqrt{3}) + 2 (- 1)$ .

$\frac{\sqrt{3} + 1}{4} + \frac{2 (\sqrt{3} - 1)}{8}$

Step 18.4

Cancel the common factors.

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Step 18.4.1

Factor $2$ out of $8$ .

$\frac{\sqrt{3} + 1}{4} + \frac{2 (\sqrt{3} - 1)}{2 (4)}$

Step 18.4.2

Cancel the common factor.

$\frac{\sqrt{3} + 1}{4} + \frac{2 (\sqrt{3} - 1)}{2 \cdot 4}$

Step 18.4.3

Rewrite the expression.

$\frac{\sqrt{3} + 1}{4} + \frac{\sqrt{3} - 1}{4}$

Step 19

Combine the numerators over the common denominator.

$\frac{\sqrt{3} + 1 + \sqrt{3} - 1}{4}$

Step 20

Rewrite $\frac{\sqrt{3} + 1 + \sqrt{3} - 1}{4}$ in a factored form.

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Step 20.1

Add $\sqrt{3}$ and $\sqrt{3}$ .

$\frac{1 + 2 \sqrt{3} - 1}{4}$

Step 20.2

Subtract $1$ from $1$ .

$\frac{0 + 2 \sqrt{3}}{4}$

Step 20.3

Add $0$ and $2 \sqrt{3}$ .

$\frac{2 \sqrt{3}}{4}$

Step 20.4

Reduce the expression $\frac{2 \sqrt{3}}{4}$ by cancelling the common factors.

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Step 20.4.1

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{2 (\sqrt{3})}{4}$

Step 20.4.2

Factor $2$ out of $4$ .

$\frac{2 \sqrt{3}}{2 \cdot 2}$

Step 20.4.3

Cancel the common factor.

$\frac{2 \sqrt{3}}{2 \cdot 2}$

Step 20.4.4

Rewrite the expression.

$\frac{\sqrt{3}}{2}$

Step 21

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{2}$

Decimal Form:

$0.86602540 \dots$